Calculus I - Lecture 20 - The Indefinite Integral Lecture Notes: http://www.math.ksu.edu/˜gerald/math220d/ Course Syllabus: http://www.math.ksu.edu/math220/spring-2014/indexs14.html Gerald Hoehn (based on notes by T. Cochran) April 6, 2014 Reminder: Exam 3 on Thursday, April 10. Review in Wednesday’s Lecture Practice Exam online on course homepage Recall that there are two main parts of Calculus 1. Derivatives: Measures instantaneous change 2. Integrals: Measures cumulative amounts We are now ready to begin part 2. It begins with the study of the reverse operation of taking the derivative. Definition (Antiderivative) A primitive or antiderivative of a function f (x) is function F (x) such that F 0(x) = f (x). Example: Find an antiderivative of x3, by trial and error. Solution: Initial guess: x4 (since derivation decreases the degree of a power function by 1): d x4 = 4 x3. dx d 1 1 Thus: ( x4) = (4x3) = x3. dx 4 4 d 1 Note: ( x4 7) = x3 dx 4 − 1 4 All functions F (x) = 4 x + C, C any constant, are antiderivatives. Did we find all antiderivatives? Theorem Let F (x) be an antiderivative of the function f (x) defined on (a, b). Then any antiderivative on (a, b) of f (x) is of the form F (x) + C for some constant C. Proof: Let G(x) be another antiderivative of F (x). Set H(x) = G(x) F (x). Then − H (x) = G (x) F (x) = f (x) f (x) = 0. 0 0 − 0 − We claim that H(x) must be a constant function. For, if it would be not, there exist (at least) two points x = u and x = v in (a, b) with H(u) = H(v). By the mean value theorem there exists then a 6 point x = c in (u, v) such that H(u) H(v) − = H0(c). u v − But since H(u) = H(v) this would mean H (c) = 0, a 6 0 6 contradiction. Thus H(x) = C for some constant C. This implies G(x) = F (x) + C. q.e.d. Definition (Indefinite Integral) The indefinite integral or general antiderivative f (x)dx of a function f (x) stands for all possible antiderivatives of f (x) defined R on an interval, i.e. f (x) dx = F (x) + C, where C is a constant Z and F (x) is an arbitrary antiderivative of f (x). Notation: In the expression f (x)dx, the function f (x) is called the integrand and dx is a differentialZ (in its symbolic meaning). The constant C as above is called the constant of integration. The indefinite integral should not be confused with the definite b integral a f (x) dx which we will consider next week and is defined as a limit of a sum. The symbol is a stretched S and R reminds about the Sum. We will also explain the relation between R the indefinite and the definite integral. Power Rule: The indefinite integral of a power function f (x) = xn, where n = 1 is 6 − 1 xn dx = xn+1 + C. n + 1 Z Raise the exponent by 1 and divide by the raised exponent. Example: Find the indefinite integral of the following functions: x14 a) f (x) = x13 f (x) dx = + C 14 Z x3/2 2x3/2 b) f (x) = √x= x1/2 f (x) dx = + C = + C 3/2 3 Z 2 1 3 x− 1 c) f (x) = = x− f (x) dx = + C = + C x3 2 −2x2 Z − d) f (x) = 1= x0 f (x) dx = x + C Z Table of Indefinite Integrals f (x) f (x)dx f (x) f (x)dx R 1 x + C R sec2 x tan x + C xn+1 xn + C, n = 1 n + 1 6 sec x tan x sec x + C 1 ln x + C 1 x | | ax ax + C x x ln a e e + C 1 arctan x + C 1 + x2 sin x cos x + C 1 − arcsin x + C √1 x2 cos x sin x + C − Proof by derivation. Guess and Fudge Method Example: Find an antiderivative of cos(3x). Solution: 1 Since cos xdx = sin x + C we try sin(3x) with fudge factor 3 : 1 1 1 3 sin(3x). Indeed ( 3 sin(3x))0 = 3 cos(3x) 3 = cos(3x). 1 R · So 3 sin(3x) is an antiderivative. The guess and fudge method applies to functions of the form f (ax + b), where a and b are constants. 1 f (ax + b)dx = F (ax + b) + C a where F (x) is anZ antiderivative of f (x). Example: 1 a) sin(2x π)dx = cos(2x π) + C − −2 − Z 5 3x 1 5 3x b) e − dx = e − + C −3 Z Rules for the indefinite integral 1) Constant factor rule: k f (x) dx = k f (x) dx · · Z Z Proof: (kF (x)) = k F (x). 0 · 0 2) Sum and difference rule: (f (x) g(x)) dx = f (x) dx g(x) dx ± ± Z Z Z Proof: (F (x) G(X )) = F (x) G (x). ± 0 0 ± 0 3x 1 Example: Find (e + 7x− ) dx. Z Solution: 3x 1 = e dx + 7 x− dx by rule 1) and 2) Z Z 1 = e3x + 7 ln x + C 3 | | 1 Example: Find + (3x + 7)5 dx. x 2 Z − Solution: (3x + 7)6 = ln x 2 + + C | − | 6 3 · dx Example: Find 1 + x2 Z Solution: 1 = dx 1 + x2 Z = arctan x + C Application to differential equations Example: Find a function f (x) such that f (x) = 6x(1 x) and f (0) = 1. 0 − Solution: f (x) is an antiderivative of 6x(1 x). Thus: − f (x) = 6x(1 x) dx − Z = (6x 6x2) dx − Z x2 x3 = 6 6 + C · 2 − · 3 = 3x2 2x3 + C − When x = 0: f (0) = 1 3 0 2 0 + C = 1 C = 1. ⇔ · − · ⇔ f (x) = 3x2 2x3 + 1 − Example: A body falls to the ground. During the fall, it feels a constant acceleration of g where g = 32 ft/sec2. At time t = 0 the body has the height y0 and the velocity v0. Find a formula for the the height y in terms of t. Solution: dy Let y = y(t) be the height function, v = v(t) = dt be the velocity dv function and a = a(t) = dt be the acceleration function. We have a(t) = g (downward acceleration). − Since v is an antiderivative of a(t) one has: v = g dt = g 1 dt = gt + C − − − Z Z v(0) = v 0 + C = v C = v 0 ⇒ 0 ⇒ 0 Thus: v = gt + v . − 0 Since y is an antiderivative of v(t) one has: t2 y = ( gt + v ) dt = g + v t + C − 0 − 2 0 Z y(0) = y0 0 + 0 + C = y0 C = y0 t2 ⇒ ⇒ Thus: y = g 2 + v0t + y0..